psubclsubN - Mathbox for Norm Megill

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Theorem psubclsubN 39927

Description: A closed projective subspace is a projective subspace. (Contributed by NM, 23-Jan-2012.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
psubclsub.s	⊢ 𝑆 = (PSubSp‘𝐾)
psubclsub.c	⊢ 𝐶 = (PSubCl‘𝐾)

**Assertion**
Ref	Expression
psubclsubN	⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

**Proof of Theorem psubclsubN**
Step	Hyp	Ref	Expression
1		eqid 2729	. . 3 ⊢ (⊥_𝑃‘𝐾) = (⊥_𝑃‘𝐾)
2		psubclsub.c	. . 3 ⊢ 𝐶 = (PSubCl‘𝐾)
3	1, 2	psubcli2N 39926	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋)
4		eqid 2729	. . . . . . 7 ⊢ (Atoms‘𝐾) = (Atoms‘𝐾)
5	4, 1, 2	psubcliN 39925	. . . . . 6 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → (𝑋 ⊆ (Atoms‘𝐾) ∧ ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) = 𝑋))
6	5	simpld 494	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ⊆ (Atoms‘𝐾))
7		psubclsub.s	. . . . . 6 ⊢ 𝑆 = (PSubSp‘𝐾)
8	4, 7, 1	polsubN 39894	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
9	6, 8	syldan 591	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆)
10	4, 7	psubssat 39741	. . . 4 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ∈ 𝑆) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
11	9, 10	syldan 591	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾))
12	4, 7, 1	polsubN 39894	. . 3 ⊢ ((𝐾 ∈ HL ∧ ((⊥_𝑃‘𝐾)‘𝑋) ⊆ (Atoms‘𝐾)) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
13	11, 12	syldan 591	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → ((⊥_𝑃‘𝐾)‘((⊥_𝑃‘𝐾)‘𝑋)) ∈ 𝑆)
14	3, 13	eqeltrrd 2829	1 ⊢ ((𝐾 ∈ HL ∧ 𝑋 ∈ 𝐶) → 𝑋 ∈ 𝑆)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ⊆ wss 3911 ‘cfv 6499 Atomscatm 39249 HLchlt 39336 PSubSpcpsubsp 39483 ⊥_𝑃cpolN 39889 PSubClcpscN 39921

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5229 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-iin 4954 df-br 5103 df-opab 5165 df-mpt 5184 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-proset 18235 df-poset 18254 df-lub 18285 df-glb 18286 df-join 18287 df-meet 18288 df-p1 18365 df-lat 18373 df-clat 18440 df-oposet 39162 df-ol 39164 df-oml 39165 df-ats 39253 df-atl 39284 df-cvlat 39308 df-hlat 39337 df-psubsp 39490 df-pmap 39491 df-polarityN 39890 df-psubclN 39922

This theorem is referenced by: pclfinclN 39937

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